"Electromagnetics"의 두 판 사이의 차이
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| 28번째 줄: | 28번째 줄: | ||
| − | + | ==Lagrangian formulation== | |
* Lagrangian for a charged particle in an electromagnetic field<br><math>L=T-V</math><br><math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br> | * Lagrangian for a charged particle in an electromagnetic field<br><math>L=T-V</math><br><math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br> | ||
| 43번째 줄: | 43번째 줄: | ||
| − | + | ==Hamiltonian formulation== | |
* total energy of a charge particle in an electromagnetic field<br><math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math><br> | * total energy of a charge particle in an electromagnetic field<br><math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math><br> | ||
| 53번째 줄: | 53번째 줄: | ||
| − | + | ==force on a particle== | |
* force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math> | * force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math> | ||
| 65번째 줄: | 65번째 줄: | ||
| − | + | ==메모== | |
* [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]<br> | * [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]<br> | ||
| 83번째 줄: | 83번째 줄: | ||
| − | + | ==encyclopedia== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 97번째 줄: | 97번째 줄: | ||
| − | + | ==books== | |
ELECTROMAGNETIC THEORY AND COMPUTATION | ELECTROMAGNETIC THEORY AND COMPUTATION | ||
2012년 10월 28일 (일) 17:16 판
gauge invariance
- the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field
- the electromagnetism is a gauge field theory with structure group U(1)
Lorentz force
- almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities
- Lorentz force is a rare example of velocity dependent force
polarization of light
- has two possibilites
- what does this mean?
Lagrangian formulation
- Lagrangian for a charged particle in an electromagnetic field
\(L=T-V\)
\(L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}\) - action
\(S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\) - Euler-Lagrange equations
\(p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}\)
\(F_{i}=\frac{\partial{L}}{\partial{{q}^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}}}\) - equation of motion
\(\dot{p}=F\) Therefore we get
\(m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}\). This is what we call the Lorentz force law. - force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)
Hamiltonian formulation
- total energy of a charge particle in an electromagnetic field
\(E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi\) - replace the momentum with the canonical momentum
- similar to covariant derivative
- similar to covariant derivative
force on a particle
- force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)
메모
- http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
- Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Classical_electromagnetism
- http://en.wikipedia.org/wiki/Maxwell's_equations
- http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations
- http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
- http://en.wikipedia.org/wiki/electrical_current
- http://en.wikipedia.org/wiki/Four-current
books
ELECTROMAGNETIC THEORY AND COMPUTATION